# L11n158

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n158 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v-u^2+u v^4-u v^3+u v^2-u v+u-v^4+2 v^3}{u v^2}$ (db) Jones polynomial $-\frac{4}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^3+a^3 z-a z$ (db) Kauffman polynomial $-z^7 a^9+5 z^5 a^9-7 z^3 a^9+3 z a^9-2 z^8 a^8+10 z^6 a^8-13 z^4 a^8+4 z^2 a^8-z^9 a^7+3 z^7 a^7+2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -3 z^8 a^6+14 z^6 a^6-17 z^4 a^6+7 z^2 a^6-a^6-z^9 a^5+4 z^7 a^5-4 z^5 a^5+4 z^3 a^5-3 z a^5+a^5 z^{-1} -z^8 a^4+4 z^6 a^4-4 z^4 a^4+2 z^2 a^4-z^5 a^3+2 z^3 a^3-z^2 a^2-z a$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-10χ
0        11
-2       21-1
-4      1  1
-6     22  0
-8    21   1
-10   12    1
-12  22     0
-14 12      1
-16 1       -1
-181        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.